# Geometric Continuity of Parametric Curves

### Brian A. Barsky and Anthony D. DeRose

###
EECS Department

University of California, Berkeley

Technical Report No. UCB/CSD-84-205

October 1984

### http://www.eecs.berkeley.edu/Pubs/TechRpts/1984/CSD-84-205.pdf

Parametric spline curves are typically constructed so that the first
*n* parametric derivatives agree where the curve segments abut. This type of continuity condition has become known as
*C^n* or
*n*th order parametric continuity. We show that the use of parametric continuity disallows many parametrizations which generate geometrically smooth curves.

We define *n*th order geometric continuity (*G^n*), develop constraint equations that are necessary and sufficient for geometric continuity of curves, and show that geometric continuity is a relaxed form of parametric continuity. *G^n* continuity provides for the introduction of *n* quantities known as shape parameters which can be made available to a designer in a computer aided design environment to modify the shape of curves without moving control vertices. Several applications of the theory are discussed, along with topics of future research.

BibTeX citation:

@techreport{Barsky:CSD-84-205, Author = {Barsky, Brian A. and DeRose, Anthony D.}, Title = {Geometric Continuity of Parametric Curves}, Institution = {EECS Department, University of California, Berkeley}, Year = {1984}, Month = {Oct}, URL = {http://www.eecs.berkeley.edu/Pubs/TechRpts/1984/5752.html}, Number = {UCB/CSD-84-205}, Abstract = {Parametric spline curves are typically constructed so that the first <i>n</i> parametric derivatives agree where the curve segments abut. This type of continuity condition has become known as <i>C^n</i> or <i>n</i>th order parametric continuity. We show that the use of parametric continuity disallows many parametrizations which generate geometrically smooth curves. <p> We define <i>n</i>th order geometric continuity (<i>G^n</i>), develop constraint equations that are necessary and sufficient for geometric continuity of curves, and show that geometric continuity is a relaxed form of parametric continuity. <i>G^n</i> continuity provides for the introduction of <i>n</i> quantities known as shape parameters which can be made available to a designer in a computer aided design environment to modify the shape of curves without moving control vertices. Several applications of the theory are discussed, along with topics of future research.} }

EndNote citation:

%0 Report %A Barsky, Brian A. %A DeRose, Anthony D. %T Geometric Continuity of Parametric Curves %I EECS Department, University of California, Berkeley %D 1984 %@ UCB/CSD-84-205 %U http://www.eecs.berkeley.edu/Pubs/TechRpts/1984/5752.html %F Barsky:CSD-84-205